2007 Mathematics

Intermediate 2 – Units 1, 2 and 3 Paper 2

Finalised Marking Instructions The Scottish Qualifications Authority 2007 The information in this publication may be reproduced to support SQA qualifications only on a non-commercial basis. If it is to be used for any other purposes written permission must be obtained from the Assessment Materials Team, Dalkeith. Where the publication includes materials from sources other than SQA (secondary copyright), this material should only be reproduced for the purposes of examination or assessment. If it needs to be reproduced for any other purpose it is the centre's responsibility to obtain the necessary copyright clearance. SQA's Assessment Materials Team at Dalkeith may be able to direct you to the secondary sources. These Marking Instructions have been prepared by Examination Teams for use by SQA Appointed Markers when marking External Course Assessments. This publication must not be reproduced for commercial or trade purposes.

Page 2

General Marking Principles These principles describe the approach to be taken when marking Intermediate 2 Mathematics papers. For more detailed guidance please refer to the notes which are included with the Marking Instructions. 1 Marks must be assigned in accordance with the Marking Instructions. The main principle in

marking scripts is to give credit for the skills demonstrated and the criteria met. Failure to have the correct method may not preclude a candidate gaining credit for the calculations involved or for the communication of the answer.

2 The answer to one part of a question, even if incorrect, must be accepted as a basis for subsequent

dependent parts of the question. Full marks in the dependent part(s) may be awarded provided the question is not simplified.

3 The following should not be penalised:

• working subsequent to a correct answer (unless it provides firm evidence that the requirements of the question have not been met)

• omission or misuse of units (unless marks have been specifically allocated for the purpose in the marking scheme)

• bad form, eg sin x° = 0.5 = 30° • legitimate variation in numerical values / algebraic expressions.

4 Solutions which seem unlikely to include anything of relevance must nevertheless be followed

through. Candidates still have the opportunity of gaining one mark or more provided the solution satisfies the criteria for the mark(s).

5 Full credit should only be given where the solution contains appropriate working. Where the

correct answer may be obtained by inspection or mentally, credit may be given, but reference to this will be made in the Marking Instructions.

6 In general markers will only be able to give credit for answers if working is shown. A wrong

answer without working receives no credit unless specifically mentioned in the Marking Instructions. The rubric on the outside of the question papers emphasises that working must be shown.

7 Sometimes the method to be used in a particular question is explicitly stated; no credit should be

given where a candidate obtains the correct answer by an alternative method. 8 Where the method to be used in a particular question is not explicitly stated, full credit must be

given for alternative methods which produce the correct answer. 9 Do not penalise the same error twice in the same question. 10 Do not penalise a transcription error unless the question has been simplified as a result. 11 Do not penalise inadvertent use of radians in trigonometry questions, provided their use is

consistent within the question.

Practical Details The Marking Instructions should be regarded as a working document and have been developed and expanded on the basis of candidates' responses to a particular paper. While the guiding principles of assessment remain constant, details can change depending on the content of a particular examination paper in a given year. 1 Each mark awarded in a question is referenced to one criterion in the marking scheme by means of

a bullet point. 2 Where a candidate has scored zero marks for any question attempted, "0" should be shown against

the answer in the place in the margin. 3 Where a marker wishes to indicate how s/he has awarded marks, the following should be used: (a) Correct working should be ticked, . (b) Where working subsequent to an error is followed through, if otherwise correct and can be

awarded marks, it should be marked with a crossed tick, . (c) Each error should be underlined at the point in the working where it first occurs. 4 Do not write any comments, words or acronyms on the scripts.

Page 3

Page 4

Mathematics Intermediate 2: Paper 2, Units 1, 2 and 3 Question

No Marking Scheme

Give 1 mark for each • Illustrations of evidence for awarding

a mark at each • 1

Ans: £30 405 •1 strategy: know how to increase by 2.3% •2 strategy: know how to calculate expected wage •3 process: carry out calculations correctly within a valid strategy

•1 1·023 •2 28 400 × 1·0233 •3 30 405 3 marks

NOTES: 1 For an answer of 30 405 without working award 3/3 2 For an answer of 30 405·01 or 30 405·02 with or without working award 2/3 3 Where an incorrect % is used, the working must be followed through to give the possibility of awarding 2/3 eg an answer of £52 849 (=28 400 × 1·233), with working award 2/3 4 For an answer of 87 160 or 87 159·60 (28 400 × 1·023 × 3), with working award 1/3 5 For an answer of 30 360 (28 400 + 28 400 × 0·023 × 3), with working award 1/3 6 For an answer of 1960 (28 400 × 0·023 × 3) award 0/3 2

Ans: 21.6 cm •1 strategy: express sector as fraction of a circle •2 strategy: know how to find length of arc •3 process: correctly calculate length of arc

•1 118/360 •2 118/360 × π × 2 × 10·5 •3 21·6 3 marks

NOTES: 1 Accept variations in π, disregard premature or incorrect rounding of 118/360 2 For 118/360 × π × 10·52 leading to 113·5 award 2/3 3 For the award of the final mark, calculations must involve π and be of equivalent difficulty

Page 5

Question

No Marking Scheme

Give 1 mark for each • Illustrations of evidence for awarding

a mark at each • 3 (a)

Ans: Boys’ data, with valid reason •1 interpret: select correct data set, with

valid reason

•1 Boys’ data, with valid reason 1 mark

NOTES: (b)

Ans: (i) 58 (ii) 52 (iii) 76 •1 process: state median •2 process: state lower quartile •3 process: state upper quartile

•1 58 •2 52 •3 76 3 marks

NOTES: 1 The first mark is available only where the median is consistent with the answer to part (a) eg Possible answers For (a) Girls’ data and (b) 56, 53, 63 award part (a) 0/1 part (b) 3/3 For (a) Girls’ data and (b) 58, 52, 76 award part (a) 0/1 part (b) 2/3 For (a) Boys’ data (with reason) and (b) 56, 53, 63 award part (a) 1/1 part (b) 2/3 2 An incorrect answer for the median must be followed through with the possibility of awarding full marks for parts (ii) and (iii)

Question

No Marking Scheme

Give 1 mark for each • Illustrations of evidence for awarding

a mark at each • (c)

Ans:

30 50 70 90

31 52 58 8876

•1 communicate correct end points •2 communicate: correct box

•1 end points at 31and 88 •2 box showing Q1, Q2, Q3 2 marks

NOTES: Incorrect answers in part (b) must be followed through to give the possibility of awarding 2/2 (d)

Ans: The girls’ results are more widely spread than the boys’ •1 communicate: valid comment about the spread of data

•1 comment

1 mark

NOTES:

Page 6

Question

No Marking Scheme

Give 1 mark for each • Illustrations of evidence for awarding

a mark at each • 4 (a)

Ans: 154° •1 process: calculate angle MTO •2 process: calculate angle MOT

•1 13° •2 154°

2 marks

NOTES:

1 Angle MTO may not be explicitly stated; it may be marked in a diagram and can be awarded the first mark

2 A correct answer, without working award 2/2 (b)

Ans: 15·6 cm •1 strategy: know to use cosine rule, sine rule or equivalent •2 process: correctly apply the cosine rule, sine rule or equivalent

•3 process: calculate MT

•1 evidence •2 += 22 8MT 8 ocos1548822 ×××−

or oo sin138

sin154MT

=

•3 15·6 cm

3 marks

NOTES:

1 Disregard errors due to premature rounding

2 Where ∠ is found to be 90° leading to an answer of 11·3, with working award 1/3 MOT

3 Where ∠ is found to be 154º, leading to an answer of 11·3 award 0/3 MOT

Page 7

Question

No Marking Scheme

Give 1 mark for each • Illustrations of evidence for awarding

a mark at each • 5

Ans: 5400 cubic centimetres •1 strategy: know how to calculate volume •2 process: substitute correctly into

formula •3 process: substitute correctly into formula •4 process: calculate volume correctly •5 process: round answer to 2 significant

figures

•1 evidence of difference in volume

of two cones

•2 241531 2 ×××π ( )5655

•3 85

31 2 ×××π ( )209

•4 5445·43 •5 5400 5 marks

NOTES:

1 Accept variations in π

2 The final mark is available for rounding an answer correct to 2 significant figures. Where the answer requires no rounding, the final mark cannot be awarded

3 For use of πr2h, the second, third and fifth marks are available Common wrong answers

5200

×××−××× 165π

312415π

31 22 award 4/5 ( × )

3600

×××−××× 85π

311651π

31 22 award 4/5 ( × )

1900

×××−××× 1651π

312415π

31 22 award 4/5 ( × )

16000 ( )85π2415π 22 ××−×× award 3/5 (× × ) 6

Ans: D is correct •1 process: state the correct letter

•1 D

1 mark

NOTES:

Page 8

Question

No Marking Scheme

Give 1 mark for each • Illustrations of evidence for awarding

a mark at each • 7 (a)

Ans: ( )( 332 − )+ xx •1 process: start to factorise •2 process: complete factorisation

•1 ( )92 2 −x •2 ( )( )332 −+ xx

2 marks

NOTES: For the following answers award 1/2 ( )92 2 −x ( )( 362 −+ xx )

) ( )( 362 +− xx (b)

Ans: 1252

−+

xx

•1 process: correctly simplify fraction

•1 1252

−+

xx

1 mark

NOTES: 1 For working subsequent to a correct answer award 0/1

Page 9

Page 10

Question No

Marking Scheme Give 1 mark for each •

Illustrations of evidence for awarding a mark at each •

8

Ans: x = − 0·7, 73 ⋅=x •1 strategy: know to use quadratic formula •2 process: substitute correctly into quadratic formula •3 process: calculate b2 − 4ac •4 process: state both values of x correct to one decimal place

•1 evidence

•2 ( )

2252466 2

×−××−−±

•3 76 •4 − 0·7, 3·7

Method 2 − possible graphical solutions •1 strategy: know to graph 562 2 −−= xxy •2 communicate: indicate position of roots •3 communicate: state first root correct to 1 decimal place •4 communicate: state second root correct to 1 decimal place

5•1 62 2 −−= xxy

•2 562 2 −−= xxy

•3 − 0·7 •4 3·7

4 marks

NOTES: 1 Where b2 − 4ac is calculated incorrectly, the final mark is available only if b2 − 4ac > 0

2 For a correct answer without working award 0/4 3 The final mark is available only when the answer requires rounding

2nd root

1st root

Question

No Marking Scheme

Give 1 mark for each • Illustrations of evidence for awarding

a mark at each • 9

Ans: 13.4 metres •1 strategy: know to find AC or BC •2 process: correct application of sine rule in triangle ABC •3 process: calculate AC or BC correctly •4 strategy: know to use right-angled trig to calculate height of block of flats or other valid strategy •5 process: calculate height of block of flats

•1 evidence of use of sine rule in triangle ABC

•2 osin38BC =

o96sin

30

or oo sin96

30

sin46=

AC

•3 BC = 18·6 m or AC = 21·7 m

•4 osin46618

h=

⋅

or osin38721=

⋅h

•5 13·4 metres

5 marks

NOTES:

1 Disregard errors due to premature rounding provided there is evidence

2 Variations in answers for a value of AC or BC or a wrong value of AC or BC must be accepted as a basis of calculating the height of triangle ABC

3 For a correct answer without working award 0/5

4 Answer obtained by a scale drawing •1 strategy: know to use scale drawing

•1

•2 process:

•3 process: •4 process: •5 process:

evidence of appropriate scale clearly stated

draw AB consistent with chosen scale

•5 m

complete triangle ACB and indicate height

measure angles of (38 ± 2)º and (46 ± 2)º

calculate height of triangle ACB correctly

Page 11

h = (13·4 ± 0·3)

Question

No Marking Scheme

Give 1 mark for each • Illustrations of evidence for awarding

a mark at each • 10

Ans: 4

5 p

•1 strategy: know how to start the division calculation •2 process: continue process •3 process: express fraction in simplest form

•1 8

5 2p ×p2

•2 pp

810 2

•3 4

5p

3 marks

NOTES: 1 A correct answer, without working award 3/3 2 An incorrect answer, without working award 0/3 11

Ans: nkpm =

•1 process: start to rearrange the formula •2 process: continue the process •3 process: make m the subject

•1 kp = m2n

•2 nkpm2 =

•3 nkpm =

3 marks

NOTES: 1 For a correct answer without working award 3/3

2 The second mark is available for division by n 3 The third mark is available for taking the square root of an expression for m2

4 For an answer of nkp

, with or without working award 2/3

Page 12

Question

No Marking Scheme

Give 1 mark for each • Illustrations of evidence for awarding

a mark at each • 12

Ans: 1/m3 •1 process: simplify expression •2 process: express with a positive power

•1 m-3 •2 1/m3

2 marks

NOTES: 13

Ans: x = 58 and 238 •1 process: solve equation for tan x° •2 process: find one value of x •3 process: find second value of x

•1 tan x° = 58

•2 x = 58 •3 x = 238

3 marks

NOTES:

1 Where tan x° is calculated incorrectly, the working must be followed through with the possibility of awarding 2/3

2 Where a graphical solution has been used, the first mark is available for indicating what graph is drawn and where the values occur 3 For a correct answer arrived at by trial and improvement, only the second and third marks are available 4 For a correct answer without working award 0/3

Page 13

Question

No Marking Scheme

Give 1 mark for each • Illustrations of evidence for awarding

a mark at each • 14

Ans: 42.7 cm •1 strategy: marshall facts and recognise

right-angle •2 strategy: use Pythagoras’ theorem or

equivalent •3 process: all calculations correct, within a

valid strategy

•1

2411

•2 222 1124 −=x •3 42·7

3 marks

NOTES: Common answers

giving 222 1324 −=x

24

13

leading to AB = 40·3 award 2/3

giving 222 51724 ⋅−=x leading to AB = 32·8 award 2/3

24 17·5

giving 222 3548 −=x leading to AB = 32·8 award 2/3

48 35

giving 222 2424 +=x

24 24

leading to AB = 33·9 award 0/3

TOTAL MARKS FOR PAPER 2

50

[END OF MARKING INSTRUCTIONS]

Page 14